Abstract: In this paper we consider the evolution of regular closed elastic curves immersed in . Equipping the ambient Euclidean space with a vector field and a function , we assume the energy of is smallest when the curvature of is parallel to , where is the unit vector field spanning the tangent bundle of . This leads us to consider a generalisation of the Helfrich functional , defined as the sum of the integral of and -weighted length. We primarily consider the case where is uniformly bounded in and is an affine transformation. Our first theorem is that the steepest descent -gradient flow of with smooth initial data exists for all time and subconverges to a smooth solution of the Euler-Lagrange equation for a limiting functional . We additionally perform some asymptotic analysis. In the broad class of gradient flows for which we obtain global existence and subconvergence, there exist many examples for which full convergence of the flow does not hold. This may manifest in its simplest form as solutions translating or spiralling off to infinity. We prove that if either and are constant, the derivative of is invertible and non-vanishing, or satisfy a `properness' condition, then one obtains full convergence of the flow and uniqueness of the limit. This last result strengthens a well-known theorem of Kuwert, Schätzle and Dziuk on the elastic flow of closed curves in where is constant and vanishes.

Glen WheelerAffiliation:
Otto-von-Guericke-Universität, Postfach 4120, D-39016 Magdeburg, Germany
Address at time of publication:
University of Wollongong, Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, Faculty of Engineering and Information Sciences, Northfields Avenue, Wollongong 2522, New South Wales, Australia
Email:
wheeler@ovgu.de, glenw@uow.edu.au